(1) Angle BAD= Angle DAC, DE is perpendicular to AB, DF is perpendicular to AC, so DE=DF, Angle AED= Angle AFD = 90, so triangular AEDs are all equal to triangular AFD, so AE=AF, so AD divides EF vertically.

(2)AD=BE=CF, so BD=EC=AF, and because Angle A= Angle B= Angle C, triangle AEF, triangle CEF and triangle BDE are congruent, so DE=EF=DF.

(3)AD = DC in the equilateral triangle ABC, so the angle DBC= 1/2, the angle ABC= 1/2, the angle DCB= angle CDE, the angle DCB = angle E+ angle CDE, so the angle DCB=2, so the angle E= angle DBC, so BD.